A quick delivery from Anwar Ali smashed through the grille of McClenaghan's helmet and hit him on the left eye.
He was taken to hospital for the wound to be stitched and later tweeted he had suffered a few broken bones.
A team spokesman said McClenaghan would undergo minor surgery in Auckland on Friday.
There was some good news for New Zealand supporters as the home team won the match by 70 runs.
McClenaghan and Matt Henry starred in a 73-run partnership for the ninth wicket that helped New Zealand post 280 runs in 50 overs.
Pakistan in reply were all out for 210.
The second game in the series is in Napier on Thursday.
For New Zealand, Henry Nicholls hit 82 and for Pakistan Mohammad Amir took three for 28.
Babar Azam made 62 for Pakistan with paceman Trent Boult taking four for 40.
South Africa lead by 254 runs
Hashim Amla scored an unbeaten half-century after on the fourth day of the fourth and final Test against England at SuperSport Park.
The hosts were 121 for four at lunch in their second innings. They now lead England by 254.
James Anderson triggered a mini collapse by striking twice in three balls, removing Stephen Cook caught behind for 25 and then trapping AB de Villiers leg before wicket for zero.
England lead the four match series 2-0.